# Designing Frequency-Dependent Impedances?

Discussion in 'Electronic Design' started by Diego Stutzer, Feb 19, 2004.

1. ### Diego StutzerGuest

Hi,
Every one knows, that e.g. a simple RC-parallel circuit has a
frequency-dependent impedance-characteristic (Absolute Value) - the
impedance (Abs) raises as the Frequency approaches zero. As a formula: Zin
= 1/(1/R + i w C) , where i ist the imaginary number and w the frequency.

Now the hard part. How does one create an Impedance, which decreases
"slower", for frequencies close to zero but then decreases "faster" for
higher frequencies, than the simple parallel RC-Circuit?
Is there some kind of procedure like the one for syntesizeing LC-Filters
(Butterworth, Chebychev,..)?

Simply increasing C does not really help, because this equals a factoring of
the frequency.
Increasing R does not help as well, as it seems.

I hope one of you cracks can help me out.
Diego Stutzer

2. ### Cecil MooreGuest

For emulation modeling, there exist programmable resistors,
capacitors, and inductors. Is that what you have in mind?

4. ### Rene TschaggelarGuest

The frequency behaviour of R, L and C corrsponds to differential
equations with one or the other value as parameter. Meaning
you're not completetly free. A book on filter design may help
clear some gaps up.
Basically : you can build what can be approximated as rational
polynominal in frequency space.

Rene

5. ### John JardineGuest

You need to graph out the required frequency-impedance slope then
approximate the required roll off rates using a segmented breakpoint scheme
consisting of a number of CR series sections in parallel. Essentially it's a
straight line approximation to the required Z-F curve. The CR's adding
zeroes as the frequency goes up.

Estimating the individual time constants can be irksome as each has effect
outside it's area of interest. Use a 'least-squares approximation' to obtain
a best curve fit for the number of sections involved.

It's an interesting subject but I've come across nothing out there that's of
use.

regards
john

6. ### Steve NoskoGuest

In other words YES. You use combinations of resistors and capacitors or
inductors. Understanding the concept of "poles" and "Zeroes" is one way
which allows the synthesis of such circuits. Another is the concept of
"corner Frequency".

7. ### Steve NoskoGuest

I think he is looking for slopes of less that 6 dB per octave.

8. ### BanGuest

___
o-|___|--+--------+--------+---o
10k | | |
| | |
--- --- ---
--- --- ---
|100n |10n | 3n3
.-. .-. |
| | | | |
| |15k | |10k |
'-' '-' |
| | |
o--------+--------+--------+---o
created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de
use fixed font to view

This does exactly what you want, in the beginning the slope is less than
3dB/oct. and at 10kHz it goes to 6dB/oct.
This is how to produce a pink noise that rolls off faster at the end of
range, or to make some weighted filters (dBA) etc.

ciao Ban

9. ### Guy MaconGuest

That's a great resource! See:
http://www.tech-chat.de/AAcircuit.html
http://www.tech-chat.de/aacircuit_tutorial.htm

Is there a human-generated english translation? If not,
I can do a machine translation with the usual humorous
but usable results.

10. ### John JardineGuest

[Slightly OT].These 'spread CR' things are *weird*. How else can 1Hz to 1MHz
be set with just one pot!.

,-------------------+--------------.
| | |
.-. .-. |
2k7| | | | |
| | | | |
'-' '-'220 |
| | |
| ¦ V+ |
| | |\| |
,---+---+--++---+---+--------- | ------|-\ |
| | | | | | Min.-. | >---'-o
.-. .-. .-. .-. .-. .-. | |<-----|+/ Square wave out
| | | | | | | | | | | | | | |/|
| | | | | | | | | | | | Max'-'Pot V-
'-' '-' '-' '-' '-' '-' | 10k
| | | | | | |
--- --- --- --- --- --- Comparitor
--- --- --- --- --- --- |
A| B| C| D| E| F| .-.
'---+---+---+---+---' | |680
| | |
0V '-'
A=10K:10u |
B=4k7:1u 0V
C=2k2:100n
D=1k2:100n
E=680:1n
F=330:100p

created by Andy´s ASCII-Circuit v1.24.140803 Beta www.tech-chat.de

regards
john

11. ### Richard ClarkGuest

Hi Diego,

This can be done in a number of ways employing active components (I've
just seen the drift of the thread take on this complexity).

The Bi-Quad filter comes to mind and has been around implemented with
Op-Amps for quite a while.

played with this for 20 years tho').

You could also build an MCU interfacing ADC and DAC chips or simply
step up to DSP chips for filters that are impossible to implement in
any combination of passive L-C-R combinations.

However, the Bi-Quad offers simultaneous Low Pass, Band Pass, Band
Reject, and High Pass outputs from one circuit configuration. For
playing around with, that flexibility is hard to beat.

73's
Richard Clark, KB7QHC

12. ### Max HauserGuest

"Diego Stutzer" in news:...
network synthesis problem (creating a network of components to realize a
prescribed signal response) and specifically the synthesis of a one-port, or
impedance.

At one time (when phone companies ruled the earth and computers had
conquered few signals and DSP was reserved for BIG things like the US
Perimeter Acquisition Radar at Concrete, North Dakota -- affectionately the
"PAR"), this was a popular subject in engineering schools at the
sometimes, especially with the sophisticated signal processing used today on
continuous-time signals in consumer products. A host of
applied-mathematical techniques (Foster and Cauer synthesis, Brune's
impedance-synthesis lemma, etc.) apply even to one-ports. Some of them are
highly counterintuitive. Not, in other words, a subject perfectly matched
to the contraints of brief advice on newsgroups. (Note also that
Butterworth and Chebyshev approximants are mathematical methods to approach
one group of curves out of things that naturally give you a different type
of characteristic -- "Butterworth and Chebyshev" have nothing to do with
specific circuit topologies or components). If you want to pursue it
further I could suggest investigating "network synthesis." Temes and
LaPatra had a reasonable modern (1970s) book about it. Karl Willy Wagner
started it all in 1915 by inventing filters.

Richard Clark suggested also investigating the small op-amp "biquad"
networks for designable frequency response (actually you can turn them into
one-ports, the so-called shunt-filter class, but again a bit of a subject
for a brief response). Note that technically a "bi-quad" is any network
giving a biquadratic transfer function (2nd-order numerator and denominator)
though in RC-active filters it's often applied to the closely related
Åkerberg-Mossberg and Tow-Thomas configurations. For practical info see van
Valkenberg's excellent general introductory book on filters from the 1980s.

For an accessible modern example of these small op-amp-based "biquad"
networks, look up the LTC1562 from Linear Technology, a commercial chip with
four trimmed "biquad" networks, programmable by outboard components for
applications from a few kHz to a few hundred kHz.

13. ### Peter O. BrackettGuest

Diego:

You cannot do *exactly* what you propose, but you can get arbitrarily
close to it.

The "closeness" being a function of the cost you are prepared to pay.

The closer you want to get to the desired function [curve] of impedance
versus
frequency, the more the cost [cost = total number of R-L-C elements in the
design].

Basically what you are trying to doe is very well known in the network
synthesis
literature as driving point impedance [DPI] synthesis. [e.g. Darlington's
method and other similar techniques. Darlngton's technique approaches
the problem of DPI as the synthesis of a lossless two port terminated
in an appropriate single resistance.]

Network synthesis was widely researched, studied and taught back in the
1940 - 1970 era but... today it is seldom seen, used, or taught. There are
however lots of older textbooks which cover this field in great depth.

I'll post a few such references here below for your reference.

Before you can actually perform the DPI synthesis you will first have to
find an appropriate rational polynomial function, to form the basis for your
synthesis, which approximates the impedance function [curve] you desire to
match. To obtain such a rational polynomial you will have to solve an
appropriate approximation problem.

Approximation theory and the techniques for doing this with rational
polynomial
are a whole 'nother problem, and other than a few simple graphical straight
line
segment tricks, will usually require the use of a computer with an
appropriate
algorithm, such as Remez second method, which you may have to write
yourself!

Check out the following classic texts on network synthesis for a complete
run down on what you need to do to accomplish your objective:

1.) Ernst A. Guillemin, "Synthesis of Passive Networks", John Wiley & Sons,
NY, 1957. [LC# 57-8886. On technical library shelves at LCShelf
Call # TK3226.G84. See Chapters 3, 4, 9, 10 which cover the DPI synthesis
in detail, and Chapter 14 which covers the approximation problem.]

2.) Norman Balabanian, "Network Syntheis" Prentice-Hall, Englewood
Cliffs, NJ 1958. [LC# 58-11650. On technical library shelves at LCC
Shelf Call # TK3226.B26. See Chapters 2 & 3 for DPI and Chapter
9 for the approximation problem.]

3.) Louis Weinberg, "Network Analysis and Synthesis", McGraw-Hill,
New York, 1962. [LC# 61-16969. On technical library shelves at
LC Shelf Call # TK3226.W395. See Chapter's 9 & 10 for DPI
synthesis and Chapter 11 for the approximation problem]

One does not have to realize such designs with purely passive RLC
networks and, in appropriate frequency ranges, they can often be
synthesized with active RC networks [R, C and Op-Amps] by
appropriate transformations of the passive synthesis results.

See for instance...

4.) Adel S. Sedra and Peter O. Brackett, "Filter Theory and Design:
Active and Passive", Matrix Publishers, Portland, OR, 1978. [LC #
76-39745. On technical library shelves at LCC Shelf Call #
TK7872.F5S42.]

Also, and I have done this myself a couple of times for special low
frequency
applications, one can match the analog driving point impedance through
an appropriate Op-Amp reflectometer circuit to a combination analog
to digital A/D and digital to analog converter D/A and perform/emulate
the DPI synthesis in real time using digital signal procssing [DSP]
techniques. Basically to use the A/D - D/A digital technique to emulate
the desired DPI you will have to solve the same synthesis and approximation
problems mentioned above but under a suitable *warping* of the real
frequency
axis.

Hope that all helps... and good luck

14. ### Jim ThompsonGuest

On Sun, 22 Feb 2004 22:34:02 GMT, "Peter O. Brackett"

[snip]
[snip]

Gawwwwd! You list makes me feel really old. Not only do I recognize
all your references, I knew "Ernie" personally... but I had Harry B.
Lee as my instructor, since I was in VI-B.

...Jim Thompson

15. ### Peter O. BrackettGuest

Jim:

[snip]
[snip]

Heh, heh... No one could "splain" driving point synthesis like "Ernie".

But... Time moves on...

Y. W. Lee, I have met, Harry B. Lee?

Best Regards,

--
Peter K1PO
Consultant - Signal Processing and Analog Electronics
Indialantic By-the-Sea, FL.

16. ### Jim ThompsonGuest

Think War of Independence, this guy was a descendent of "Light-Horse"
Harry Lee ;-)

Harry B. Lee was, IMNSHO, better than Ernie. Everything Ernie did was
1H, 1F and 1 ohm... Harry was a realist, and I still have my notes...
I don't think anyone could have taught nodal and loop analysis better;

...Jim Thompson

17. ### Peter O. BrackettGuest

Jim:

[snip]
[snip]

And of course... "pliers" and "soldering iron"...

[snip]
Harry was a realist, and I still have my notes...
[snip]

As I said, I never had the pleasure, but I'm sure he was a grea teacher...

[snip]
[snip]

Hmmm... how about the "Method of False Assumption"?